I don't understand the concept of unit cell and the number of atoms per unit cell in a cubic lattice also the calculations for the number of atoms. For example in the $\ce{fcc}$ lattice, number of atoms per unit cell is:
$$8\cdot\frac{1}{8} + 6\cdot\frac{1}{2}=4$$
- what does the 2 and 8 in the denominator stand for?
- also 4?
Answer
The denominator signifies the number of cubes that are needed to completely encompass the whole point. For example, a corner point can be thought of as a center of 8 whole cubes, while a face centre is enclosed by 2 cubes and an edge center by 4. Hence, only 1/8 of a corner atom is in a specific unit cell and so on and so forth.
Consequently, the total number of atoms in a unit cell (say a FCC) would be equal to -
(no of corners)(fraction of corner in the unit cell) = 8(1/8)
plus
(no of face centers)(fraction of face center in the unit cell) = 6(1/2)
which equals to 4
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